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Theorem dfoprab2 6654
Description: Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)
Assertion
Ref Expression
dfoprab2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤   𝜑,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem dfoprab2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 excom 2039 . . . 4 (∃𝑧𝑤𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤𝑧𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2 exrot4 2043 . . . . 5 (∃𝑧𝑤𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥𝑦𝑧𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3 opeq1 4370 . . . . . . . . . . . 12 (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
43eqeq2d 2631 . . . . . . . . . . 11 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑣 = ⟨𝑤, 𝑧⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
54pm5.32ri 669 . . . . . . . . . 10 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
65anbi1i 730 . . . . . . . . 9 (((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑))
7 anass 680 . . . . . . . . 9 (((𝑣 = ⟨𝑤, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
8 an32 838 . . . . . . . . 9 (((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ∧ 𝜑) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
96, 7, 83bitr3i 290 . . . . . . . 8 ((𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
109exbii 1771 . . . . . . 7 (∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩))
11 opex 4893 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
1211isseti 3195 . . . . . . . 8 𝑤 𝑤 = ⟨𝑥, 𝑦
13 19.42v 1915 . . . . . . . 8 (∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩))
1412, 13mpbiran2 953 . . . . . . 7 (∃𝑤((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ∧ 𝑤 = ⟨𝑥, 𝑦⟩) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
1510, 14bitri 264 . . . . . 6 (∃𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
16153exbii 1773 . . . . 5 (∃𝑥𝑦𝑧𝑤(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
172, 16bitri 264 . . . 4 (∃𝑧𝑤𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
18 19.42vv 1917 . . . . 5 (∃𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
19182exbii 1772 . . . 4 (∃𝑤𝑧𝑥𝑦(𝑣 = ⟨𝑤, 𝑧⟩ ∧ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
201, 17, 193bitr3i 290 . . 3 (∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2120abbii 2736 . 2 {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))}
22 df-oprab 6608 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
23 df-opab 4674 . 2 {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑤𝑧(𝑣 = ⟨𝑤, 𝑧⟩ ∧ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))}
2421, 22, 233eqtr4i 2653 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  {cab 2607  cop 4154  {copab 4672  {coprab 6605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-opab 4674  df-oprab 6608
This theorem is referenced by:  reloprab  6655  oprabv  6656  cbvoprab1  6680  cbvoprab12  6682  cbvoprab3  6684  dmoprab  6694  rnoprab  6696  ssoprab2i  6702  mpt2mptx  6704  resoprab  6709  funoprabg  6712  elrnmpt2res  6727  ov6g  6751  dfoprab3s  7168  xpcomco  7994  omxpenlem  8005  nvss  27297  mpt2mptxf  29320  bj-dfmpt2a  32708  mpt2mptx2  41401
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